RIGHT/LEFT/MID Functions

=MID(A2,FIND("""",A2)+1,FIND(">",A12)-FIND("""",A2)-2)

This should do the job for you.

Your original formula could also be modified to do this to:
=LEFT(RIGHT(A2,LEN(A2)-FIND("""",A2)),LEN(RIGHT(A2,LEN(A2)-FIND("""",A2)))-2)

To understnad formulas you start from the inside out. So for cheal2's formula =left(right(a2,9),7) you need to fist understand:

right(a2,9)
This will display text from A2 9 characters from the right = nolines">
Then the left(X,7) part kicks in. Note X has been evaluated to nolines"> so you can say the formula is now =left(nolines">,7)

This takes the text nolines"> and shows the first 7 characters = nolines
The only disadvantage to this method is that it assumes you always have the same text layout in A2.

My formula (which I believe can be tidied up but i'm not 100% with text manipulation) will return any text within the " bracket (assuming that the second " is followed by a >).

Breakdown:
=MID(A2,FIND("""",A2)+1,FIND(">",A12)-FIND("""",A2)-2) whole formula

FIND("""",A2)+1 returns the location on the first " in A2 and adds 1. Adding 1 is used because you don't want to include the " in your output.

This gives:
=MID(A2,12,FIND(">",A12)-12-2)

FIND(">",A12) is used to find the location of > so the formula can now be looked at as:
=MID(A2,12,20-12-2)

The mid formula needs to know the cell to look at (A2) the first character location you want to start at (12) and the number of characters from the start location to show (20-12-2).
The 20 is the location of > in A2, -12 is the start number and -2 is because you want to exclude the "> from the end.

The advantage of this formula is that you can enter any text into A2 and get the required output, eg:

<table id="nolines"> would become nolines
<something else instead ="resulting text to show"> would become resulting text to show

I hope that makes sense, I always find it hard to explain how things work!